Vtyjkyuk

So the question is how do we know if an argument without premises is valid.

First of all, how would that go? I mean, what would an argument without premises look like (in terms of propositional logic)? Would there just be a conclusion? I also read that in case of a tautology, that would be a valid argument and I simply don't understand how there even can be a truth table created if there are no premises.

Also, why is $p$ or not $p$ an argument without premises? Isn't $p$ itself a premise?

Excuse my probably very simple questions, I'm very new to propositional logic or rather discrete math as a whole.

I simply don't understand how there even can be a truth table created if there are no premises.

With $n$ statements, there are $2^n$ ways to conjoin each statement or its negation with the others, and $2^2^n$ to disjoin these i.e. $2^2^n$ truth functions. With no premises, set $n=0$ so there are $2$ truth functions, true and false (or if you prefer, tautology and contradiction). The argument will just be a conclusion, and is valid iff the conclusion is tautological.

Also, why is $p$ or not $p$ an argument without premises? Isn't $p$ itself a premise?

To assume $p$ would in general be a premise, but that's not what you're doing. The simplest explanation is a comment by Ludwig Wittgenstein: "For example, I know nothing about the weather when I know that it is either raining or not raining."

An argument without premises is a single sentence : the conclusion.

A sentence is valid

if and only if it is true under every possible interpretation of the language. In propositional logic, they are tautologies.

Regarding truth table, there is no issue with a truth table for a single formula.

Tautologies are exactly those formulas whose rightmost column in the truth table shows only the value TRUE.